I survived flying SYD-LAX-JFK in Economy on an American airline before, so it can't be that bad! The 14 hours will be a bit painful, but you'll make it. I can't remember it being much more turbulent than any other flights I've taken. Exit row is worth it if you can get it, and you'll want to bring some entertainment + noise-cancelling headphones + earplugs. Given you've got a few days in LA, your flight to NYC will be just fine - I managed it with a whole 2 hours of connecting time in LAX, but having a few days in LA is probably nicer.
NYC is amazing - I went there in winter, the weather was cold but the city was just as alive as it always is. And you've got plenty of places to visit in the north-east - I spent a week in DC which was absolutely fantastic.
The flights are a bit painful, but absolutely manageable, and America's well worth seeing, so you'll regret it if you don't do it!